I am having some difficulty finding a closed-form for this
$\displaystyle\sum_{x=1}^{L}(L+1-xa)(L+1-xb)(2^{x-1}-1)$ (assume $L$, $a$, and $b$ are known)
Or does it not exist?
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By multiplying out, we can reduce the problem to sums of the form (i) $\sum 2^{x-1}$ (geometric series, familiar formula), (ii) $\sum x2^{x-1}$, and (iii) $\sum x^22^{x-1}$. (There will also be terms of the type $\sum x$ and $\sum x^2$, that are settled by standard formulas.)
The types (ii) and (iii) have often come up om MSE. There are many approaches. For example, consider $F(n)=\sum_1^n x 2^{x-1}$. Thus $$F(n)=1\cdot 2^0 +2\cdot 2^1+3\cdot 2^2+\cdots +n\cdot n2^{n-1}.$$ Multiply the above expression for $F(n)$ by $2$. We get $$2F(n)=1\cdot 2^1+2\cdot 2^2 +3\cdot 2^3+\cdots +n\cdot 2^n.$$ Subtract. We get $$2F(n)-F(n)=F(n)= -(2^0+2^1+2^2+\cdots +2^{n-1})+n2^n.$$ The part in parentheses above is a finite geometric series, with easily computed sum.
For $\sum_1^n x^2 2^{x-1}$, use fundamentally the same trick, except that when we subtract we find that we need $\sum x2^{x-1}$. Conveniently, that sort of sum has just been calculated.
André Nicolas
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Do I absolutely have to multiply everything out? I had done so on Wolfram and it looked quite long and unreadable. You are saying I have to multiply it all out and then sum each piece separately? – NullOverNull Feb 22 '13 at 20:17
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See http://tinyurl.com/b9zw4nq – NullOverNull Feb 22 '13 at 20:25
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There may be shortcuts. I do not see anything that helps dramatically. – André Nicolas Feb 22 '13 at 20:43
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I did a quick calculation, got the same answer as you did, in slightly different form. There is good reason to think we did not both err. – André Nicolas Feb 22 '13 at 21:11